American Institute of Mathematical Sciences

2012, 17(3): 977-992. doi: 10.3934/dcdsb.2012.17.977

A constructive proof of the existence of a semi-conjugacy for a one dimensional map

 1 School of Mathematical and Statistical Sciences, Arizona State University, Tempe, Arizona 85287-1804, United States 2 Department of Financial and Computational Mathematics, Providence University, Taichung 43301, Taiwan

Received  March 2011 Revised  September 2011 Published  January 2012

A continuous map $f:[0,1]\rightarrow[0,1]$ is called an $n$-modal map if there is a partition $0=z_0 < z_1 < ... < z_n=1$ such that $f(z_{2i})=0$, $f(z_{2i+1})=1$ and, $f$ is (not necessarily strictly) monotone on each $[z_{i},z_{i+1}]$. It is well-known that such a map is topologically semi-conjugate to a piecewise linear map; however here we prove that the topological semi-conjugacy is unique for this class of maps; also our proof is constructive and yields a sequence of easily computable piecewise linear maps which converges uniformly to the semi-conjugacy. We also give equivalent conditions for the semi-conjugacy to be a conjugacy as in Parry's theorem. Related work was done by Fotiades and Boudourides and Banks, Dragan and Jones, who however only considered cases where a conjugacy exists. Banks, Dragan and Jones gave an algorithm to construct the conjugacy map but only for one-hump maps.
Citation: Dyi-Shing Ou, Kenneth James Palmer. A constructive proof of the existence of a semi-conjugacy for a one dimensional map. Discrete & Continuous Dynamical Systems - B, 2012, 17 (3) : 977-992. doi: 10.3934/dcdsb.2012.17.977
References:
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References:
 [1] , IEEE standard for floating-point arithmetic,, The Institute of Electrical and Electronics Engineers, (2008). [2] Ll. Alsedà, J. Llibre and M. Misurewicz, "Combinatorial Dynamics and Entropy in Dimension One,", 2nd edition, 5 (2000). [3] J. Banks, V. Dragan and A. Jones, "Chaos: A Mathematical Introduction,", Australian Mathematical Society Lecture Series, 18 (2003). [4] K. M. Brucks and H. Bruin, "Topics from One-Dimensional Dynamics,", London Mathematical Society Student Texts, 62 (2004). [5] R. L. Devaney, "An Introduction to Chaotic Dynamical Systems,", 2nd edition, (1989). [6] N. A. Fotiades and M. A. Boudourides, Topological conjugacies of piecewise monotone interval maps,, International Journal of Mathematics and Mathematical Sciences, 25 (2001), 119. doi: 10.1155/S0161171201004343. [7] P. Henrici, "Essentials of Numerical Analysis with Pocket Calculator Demonstrations,", John Wiley & Sons, (1982). [8] J. Milnor and W. Thurston, On iterated maps of the interval,, in, 1342 (1988), 1986. [9] W. Parry, Symbolic dynamics and transformations of the unit interval,, Transactions of the American Mathematical Society, 122 (1966), 368. doi: 10.1090/S0002-9947-1966-0197683-5.
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